A company has a policy of retiring company cars; this policy looks at number of miles driven, purpose of trips, style of car and other features. The distribution of the number of months in service for the fleet of cars is bell-shaped and has a mean of 53 months and a standard deviation 55 months. Using the Empirical Rule rule, what is the approximate percentage of cars that remain in service between 38 and 43 months? Do not enter the percent symbol. ans =

To use the Empirical Rule (also known as the 68-95-99.7 rule), we first need to find how many standard deviations the values 38 and 43 months are from the mean (53 months). 1. Calculate the z-scores: - For 38 months: \( z = \frac{38 - 53}{55} \approx -0.27 \) - For 43 months: \( z = \frac{43 - 53}{55} \approx -0.18 \) These z-scores are not exact multiples of the standard deviation, but we can still approximate. The Empirical Rule states that about 68% of the data falls within one standard deviation (between 48 and 58 months), and about 95% falls within two standard deviations (between 43 and 63 months). Given that both 38 and 43 months fall close to two standard deviations below the mean, we can estimate that around 95% of the cars are in service between two standard deviations from the mean. However, since 38 months is slightly outside of this range, we can refine our estimate downward. Thus, we can reasonably estimate that approximately 14% of cars (which translates to those just below the two-standard deviation mark) are likely to still be running in this range. ans = 14